how likely is contagion in financial networks?...how likely is contagion in financial networks? ......
TRANSCRIPT
OFFICE OFFINANCIAL RESEARCHU.S. DEPARTMENT OF THE TREASURY
Office of Financial Research Working Paper #0009
June 21, 2013
How Likely is Contagion in Financial Networks?
Paul Glasserman,1 and H. Peyton Young2
1 Columbia Business School, Columbia University [email protected] 2 Department of Economics, University of Oxford, and Office of Financial Research, U.S. Department of Treasury, [email protected]
The Office of Financial Research (OFR) Working Paper Series allows staff and their co-authors to disseminate preliminary research findings in a format intended to generate discussion and critical comments. Papers in the OFR Working Paper Series are works in progress and subject to revision. Views and opinions expressed are those of the authors and do not necessarily represent official OFR or Treasury positions or policy. Comments are welcome as are suggestions for improvements, and should be directed to the authors. OFR Working Papers may be quoted without additional permission.
www.treasury.gov/ofr
How Likely is Contagion in Financial Networks?
Paul Glasserman∗ and H. Peyton Young†
Abstract
Interconnections among financial institutions create potential channels for contagion and amplifica-tion of shocks to the financial system. We estimate the extent to which interconnections increaseexpected losses, with minimal information about network topology, under a wide range of shockdistributions. Expected losses from network effects are small without substantial heterogeneity inbank sizes and a high degree of reliance on interbank funding. They are also small unless shocksare magnified by some mechanism beyond simple spillover effects; these include bankruptcy costs,fire sales, and mark-to-market revaluations of assets. We illustrate the results with data on theEuropean banking system.
Keywords: systemic risk, contagion, financial network
JEL: D85, G21
We thank Thomas Noe and Alireza Tahbaz-Salehi for constructive comments on an earlier draft.The views and opinions expressed are those of the authors and do not necessarily represent official OFRor Treasury positions or policy. Comments are welcome as are suggestions for improvements and shouldbe directed to the authors. OFR Working Papers may be quoted without additional permission.
∗Columbia Business School, Columbia University, [email protected]†Department of Economics, University of Oxford, and Office of Financial Research, U.S. Treasury, pey-
1 Introduction
The interconnectedness of the modern financial system is widely viewed as having been a key contributing
factor to the recent financial crisis. Due to the complex web of links between institutions, stresses to one
part of the system can spread to others, leading to a system-wide threat to financial stability. Specific
instances include the knock-on effects of the Lehman bankruptcy, potential losses to counterparties that
would have resulted from a failure of the insurance company AIG, and more recently the exposure of
European banks to the risk of sovereign default by some European countries. These and other examples
highlight concerns that interconnectedness could pose a significant threat to the stability of the financial
system.1 Moreover there is a growing body of research that shows how this can happen in a theoretical
sense.2
Although it is intuitively clear that interconnectedness has some effect on the transmission of shocks,
it is less clear whether and how it significantly increases the likelihood and magnitude of losses compared
to a financial system that is not interconnected. In this paper we propose a general framework for
analyzing this question. There are in fact many different types of networks connecting parts of the
financial system, including networks defined through ownership hierarchies, payment systems, derivatives
contracts, brokerage relationships, and correlations in stock prices, among other examples. We focus on
the network defined by liabilities between financial institutions. These payment obligations create the
most direct channel for the spread of losses.
It turns out that one can derive general bounds on the effects of this source of interconnectedness
with almost no information about the network topology: our bounds hold independently of the degree
distribution, amount of connectivity, node centrality, average path length, and so forth. The topology-
free property of our results is one of the main contributions of this work. Moreover, we impose probability
distributions on the shocks to the nodes and show that the same bounds hold for a wide range of
distributions, including beta, exponential, normal, and many others. This robustness is important
because detailed information about interbank liabilities is often unavailable and the exact form of the
shock distributions is subject to considerable uncertainty.
To model a network of payment obligations, we build on the elegant framework of Eisenberg and
Noe (2001). The model specifies a set of nodes that represent financial institutions together with the
obligations between them. Given an initial shock to the balance sheets of one or more nodes, one can
compute a set of payments that clear the network by solving a fixed-point problem. This framework is1See, for example, Bank of England 2011, International Monetary Fund 2012, and Office of Financial Research 2012.2See in particular Allen and Gale (2000), Upper and Worms (2002), Degryse and Nguyen (2004), Goodhart, Sunirand,
and Tsomocos (2004), Elsinger, Lehar, and Summer (2006), Allen and Babus (2009), Gai and Kapadia (2010), Gai,Haldane, and Kapadia (2011), Haldane and May (2011), Upper (2011), Allen and Carletti (2011), Georg (2011), Rogersand Veraart (2012), Acemoglu, Ozdaglar, and Tahbaz-Salehi (2013), and Elliott, Golub, and Jackson (2013).
1
very useful for analyzing how losses propagate through the financial system. A concrete example would
be delinquencies in mortgage payments: if some fraction of a banks mortgages are delinquent and it has
insufficient reserves to cover the shortfall, then it will be unable to pay its creditors in full, who may
be unable to pay their creditors in full, and so forth. The original shortfall in payments can cascade
through the system, causing more and more banks to default through a domino effect. The Eisenberg-Noe
framework shows how to compute a set of payments that clear the network, and it identifies which nodes
default as a result of an initial shock to the system. The number and magnitude of such defaults depend
on the topological structure of the network, and there is now a substantial literature characterizing those
structures that tend to propagate default or alternatively that tend to dampen it (Gai and Kapadia 2010,
Gai, Haldane, and Kapadia 2011, Haldane and May, 2011, Acemoglu, Ozdaglar, and Tahbaz-Salehi 2013,
and Elliott, Golub, and Jackson 2013).
Much of this literature proceeds by examining the effects of fixed shocks applied to particular nodes
rather than fully specifying the distribution that generates the shocks. In this paper we analyze the
probability of contagion and the expected losses generated by contagion when the joint distribution of
shocks is given. We then apply the framework to answer the following questions about the impact of
network effects. First, how likely is it that a given set of banks will default due to contagion from another
node, as compared to the likelihood that they default from direct shocks to their own assets? Second,
how much does the network increase the probability and magnitude of losses compared to a situation
where there are no connections?
To compare systems with and without interconnections, we proceed as follows. First, we define our
nodes to be financial institutions that borrow and lend on a significant scale, which together with their
obligations to one another constitute the financial network. In addition, such institutions borrow and
lend to the nonfinancial sector, which is composed of investors, households, and nonfinancial firms. We
compare this system to one without connections that is constructed as follows. We remove all of the
obligations between the financial nodes while keeping their links with the nonfinancial sector unchanged.
We also keep node equity values as before by creating, for each node, a fictitious outside asset (or liability)
whose value equals the net value of the connections at that node that were removed. We then apply
the same shock distributions to both systems, with the shocks to real assets originating in the external
sector and the fictitious assets (if any) assumed to be impervious to shocks. We can ascertain how much
the network connections contribute to increased defaults and losses by comparing the outcomes in the
two systems.
One might suppose that the comparison hinges on what shock distribution we use, but this turns
out not to be the case: we show how to compute general bounds on the increased losses attributable to
network contagion that hold under a wide variety of distributions. The bounds also hold whether the
2
shocks are independent or positively associated and thus capture the possibility that institutions have
portfolios that are exposed to common shocks (see for example Caccioli et al. 2012).
Two key findings emerge from this analysis. First we compute the probability that default at a given
node causes defaults at other nodes (via network spillovers), and compare this with the probability that
all of these nodes default by direct shocks to their outside assets with no network transmission. We
derive a general formula that shows when the latter probability is larger than the former, in which case
we say that contagion is weak. A particular implication is that contagion is always weak unless there is
substantial heterogeneity in node sizes as measured by their claims outside the financial sector. More
generally, contagion will tend to be weak unless the originating node is large, highly leveraged, and –
crucially – has a relatively high proportion of its obligations to other financial institutions as opposed to
the nonfinancial sector. Second, the analysis shows that the total additional losses generated by network
spillover effects are surprisingly small under a wide range of shock distributions for plausible values of
model parameters. Both of these results are consistent with the empirical and simulation literature on
network stress testing, which finds that contagion is quite difficult to generate through the interbank
spillover of losses (Degryse and Nguyen 2004, Elsinger, Lehar, and Summer 2006, Furfine 2003, Georg
2011, Nier et al. 2007). Put differently, our results show that contagion through spillover effects becomes
most significant under the conditions described in Yellen (2013), when financial institutions inflate their
balance sheets by increasing leverage and expanding interbank claims backed by a fixed set of real assets.
These results do not imply that all forms of network contagion are unimportant; rather they show that
simple spillover or “domino” effects have only a limited impact at realistic levels of payment obligations
between banks. This leads us to examine other potential sources of contagion. The prior literature
has focused on the role of fire sales, that is, the dumping of assets on the market in order to cover
losses.3 Here we shall focus on two alternative mechanisms that are more immediate extensions of the
simple spillover mechanism in the Eisenberg-Noe model and thus allow a more immediate comparison:
bankruptcy costs and losses of confidence.
Bankruptcy costs magnify the costs associated with default both directly, through costs like legal
fees, and indirectly through delays in payments to creditors and disruptions to the provision of financial
intermediation services necessary to the real economy. We model these effects in reduced form through a
multiplier on losses when a node defaults. This approach allows us to estimate how much the probability
of contagion, and the expected losses induced by contagion, increase as a function of bankruptcy costs. A
somewhat surprising finding is that bankruptcy costs must be quite large in order to have an appreciable
impact on expected losses as they propagate through the network.
A second mechanism that we believe to be of greater importance is crises of confidence in the credit3See Shleifer and Vishny (2011) for a survey and Cifuentes, Ferrucci, and Shin (2005) for an extension of the Eisenberg-
Noe framework with fire sales.
3
quality of particular firms. If a firm’s perceived ability to pay declines for whatever reason, then so does
the market value of its liabilities. In a mark-to-market regime this reduction in value can spread to other
firms that hold these liabilities among their assets. In other words, the mere possibility (rather than the
actuality) of a default can lead to a general and widespread decline in valuations, which may in turn
trigger actual defaults through mark-to-market losses.4 This is an important phenomenon in practice:
indeed it has been estimated that mark-to-market losses from credit quality deterioration exceeded losses
from outright defaults in 2007-2009.5
We capture this idea by re-interpreting the Eisenberg-Noe framework as a valuation model rather
than as a clearing model. Declines in confidence about the ability to pay at some nodes can spread
to other nodes through a downward revaluation of their assets. This mechanism shows how a localized
crisis of confidence can lead to widespread losses of value. Our analysis suggests that this channel of
contagion is likely to be considerably more important than simple domino or spillover effects.
The rest of the paper is organized as follows. In Section 2 we present the basic Eisenberg-Noe
framework and illustrate its operation through a series of simple examples. In Section 3 we introduce
shock distributions explicitly. We then compare the probability that a given set of nodes default from
simultaneous direct shocks to their outside assets, with the probability that they default indirectly by
contagion from some other node. In Section 4 we examine the expected loss in value that is attributable
to network contagion using the comparative framework described above. We show that one can obtain
useful bounds on the losses attributable to the network with almost no knowledge of the specific network
topology and under very general assumptions about the shock distributions. In Section 5 we introduce
bankruptcy costs and show how to extend the preceding analysis to this case. Section 6 examines the
effects of a deterioration in confidence at one or more institutions, such as occurred in the 2008-09
financial crisis. We show how such a loss of confidence can spread through the entire system due to
mark-to-market declines in asset values. In the Appendix we illustrate the application of these ideas to
the European banking system using data from European Banking Authority (2011).
2 Measuring systemic risk
2.1 The Eisenberg-Noe framework
The network model proposed by Eisenberg and Noe (2001) has three basic ingredients: a set of n nodes
N = {1, 2, ..., n}, an n × n liabilities matrix P = (pij) where pij ≥ 0 represents the payment due from
4This mechanism differs from a bank run, which could also be triggered by a loss of confidence. Mark-to-market lossesspread when a lender continues to extend credit, whereas a run requires withdrawal of credit. In the seminal frameworkof Diamond and Dybvig (1983), a run is triggered by a demand for liquidity rather than a concern about credit quality.
5According to the Basel Committee on Banking Supervision, for example, roughly two thirds of losses attributed tocounterparty credit risk were due to mark-to-market losses and only about one third of losses were due to actual defaults.See http://www.bis.org/press/p110601.htm.
4
wi
bi
ci wj
wk
ijp
kip
Figure 1 Figure 1: Node i has an obligation pij to node j, a claim pkj on node k, outside assets ci, and outsideliabilities bi, for a net worth of wi.
node i to node j, pii = 0, and a vector c = (c1, c2, ..., cn) ∈ Rn+ where ci ≥ 0 represents the value
of outside assets held by node i in addition to its claims on other nodes in the network. Typically ci
consists of cash, securities, mortgages and other claims on entities outside the network. In addition each
node i may have liabilities to entities outside the network; we let bi ≥ 0 denote the sum of all such
liabilities of i, which we assume have equal priority with i’s liabilities to other nodes in the network.
The asset side of node i’s balance sheet is given by ci +∑
j 6=i pji, and the liability side is given by
pi = bi +∑
j 6=i pij . Its net worth is the difference
wi = ci +∑j 6=i
pji − pi. (1)
The notation associated with a generic node i is illustrated in Figure 1. Inside the network (indicated
by the dotted line), node i has an obligation pij to node j and a claim pki on node k. The figure also
shows node i’s outside assets ci and outside liabilities bi. The difference between total assets and total
liabilities is the node’s net worth wi.
Observe that i’s net worth is unrestricted in sign; if it is nonnegative then it corresponds to the
book value of i’s equity. We call this “book value” because it is based on the nominal or face value of
the liabilities pji, rather than on “market” values that reflect the nodes’ ability to pay. These market
values depend on other nodes’ ability to pay conditional on the realized value of their outside assets.
To be specific, let each node’s outside assets be subjected to a random shock that reduces the value of
its outside assets, and hence its net worth. These are shocks to “fundamentals” that propagate through
the network of financial obligations. Let Xi ∈ [0, ci] be a random shock that reduces the value of i’s
outside assets from ci to ci−Xi. After the shock, i’s net worth has become wi−Xi. Let F (x1, x2, ..., xn)
be the joint cumulative distribution function of these shocks; we shall consider specific classes of shock
distributions in the next section. (We use Xi to denote a random variable and xi to denote a particular
5
5
10
50
55
10
50
55
10
5
100 150
10
5 5
55
50
55
50
10
Figure 2
5
10
50
55
10
50
55
10
5
100 150
10
5 5
55
50
55
50
10 y
y
y
y
Figure 3
(a) (b)
Figure 2: Two network examples.
realization.)
To illustrate the effect of a shock, we consider the numerical example in Figure 2(a), which follows
the notational conventions of Figure 1. In particular, the central node has a net worth of 10 because it
has 150 in outside assets, 100 in outside liabilities, and 40 in liabilities to other nodes inside the network.
A shock of magnitude 10 to the outside assets erases the central node’s net worth, but leaves it with just
enough assets (140) to fully cover its liabilities. A shock of magnitude 80 leaves the central node with
assets of 70, half the value of its liabilities. Under a pro rata allocation, each liabilitity is cut in half,
so each peripheral node receives a payment of 5, which is just enough to balance each peripheral node’s
assets and liabilities. Thus, in this case, the central node defaults but the peripheral nodes do not. A
shock to the central node’s outside assets greater than 80 would reduce the value of every node’s assets
below the value of its liabilities.
Figure 2(b) provides a more complex version of this example in which a cycle of obligations of size
y runs through the peripheral nodes. To handle such cases, we need the notion of a clearing vector
introduced by Eisenberg and Noe (2001).
The relative liabilities matrix A = (aij) is the n× n matrix with entries
aij ={
pij/pi, if pi > 00, if pi = 0 (2)
Thus aij is the proportion of i’s obligations owed to node j. Since i may also owe entities in the external
sector,∑
j 6=i aij ≤ 1 for each i, that is, A is row substochastic.6
Given a shock realization x = (x1, x2, ..., xn) ≥ 0, a clearing vector p(x) ∈ Rn+ is a solution to the
system
pi(x) = pi ∧ (∑
j
pj(x)aji + ci − xi)+. (3)
6The row sums are all equal to 1 in Eisenberg and Noe (2001) because bi = 0 in their formulation.
6
As we shall subsequently show, the clearing vector is unique if the following condition holds: from every
node i there exists a chain of positive obligations to some node k that has positive obligations to the
external sector. (This amounts to saying that A has spectral radius less than 1.) We shall assume that
this condition holds throughout the remainder of the paper.
2.2 Mark-to-market values
The usual way of interpreting p(x) is that it corresponds to the payments that balance the realized assets
and liabilities at each node given that: i) debts take precedence over equity; and ii) all debts at a given
node are written down pro rata when the net assets at that node (given the payments from others), is
insufficient to meet its obligations. In the latter case the node is in default, and the default set is
D(p(x)) = {i : pi(x) < pi(x)}. (4)
However, a second (and, in our setting, preferable) way of interpreting p(x) is to see (3) as a mark-to-
market valuation of all assets following a shock to the system. The nominal value pi of node i’s liabilities
is marked down to pi(x) as a consequence of the shock x, including its impact on other nodes. As in our
discussion above of Figure 2(a), after marking-to-market, node i’s net worth is reduced from (1) to
ci − xi +∑j 6=i
pji(x)− pi(x). (5)
The reduction in net worth reflects both the direct effect of the shock component xi and the indirect
effects of the full shock vector x. Note, however, that this is a statement about values; it does not require
that the payments actually be made at the end of the period. Under this interpretation p(x) provides a
consistent re-valuation of the assets and liabilities of all the nodes when a shock x occurs.
As shown by Eisenberg and Noe, a solution to (3) can be constructed iteratively as follows. Given a
realized shock vector x define the mapping Φ : Rn+ → Rn
+ as follows:
∀i, Φi(p) = pi ∧ (∑
j
pjaji + ci − xi)+. (6)
Starting with p0 = p let
p1 = Φ(p0), p2 = Φ(p1), ... (7)
This iteration yields a monotone decreasing sequence p0 ≥ p1 ≥ p2... . Since it is bounded below it has
a limit p′, and since Φ is continuous p′ satisfies (3). Hence it is a clearing vector.
We claim that p′ is in fact the only solution to (3). Suppose by way of contradiction that there is
another clearing vector, say p′′ 6= p′. As shown by Eisenberg and Noe, the equity values of all nodes
must be the same under the two vectors, that is,
p′A + (c− x)− p′ = p′′A + (c− x)− p′′.
7
Rearranging terms it follows that
(p′′ − p′)A = p′′ − p′, where p′′ − p′ 6= 0.
This means that the matrix A has eigenvalue 1, which is impossible because under our assumptions A
has spectral radius less than 1.
3 Estimating the probability of contagion
Systemic risk can be usefully decomposed into two components: i) the probability that a given set of
nodes D will default and ii) the loss in value conditional on D being the default set. This decomposition
allows us to distinguish between two distinct phenomena: contagion and amplification. Contagion occurs
when defaults by some nodes trigger defaults by other nodes through a domino effect. Amplification
occurs when contagion stops but the losses among defaulting nodes keep escalating because of their
indebtedness to one another. Roughly speaking the first effect corresponds to a “widening” of the crisis
whereas the second corresponds to a “deepening” of the crisis. In this section we shall examine the
probability of contagion; the next section deals with the amplification of losses due to network effects.
To estimate the probability of contagion we shall obviously need to make assumptions about the
distribution of shocks. We claim, however, that we can estimate the relative probability of contagion
versus simultaneous default with virtually no information about the network structure and relatively
weak conditions on the shock distribution.
To formulate our results we shall need the following notation. Let βi = pi/(bi + pi) be the proportion
of i’s liabilities to other entities in the financial system.7 We can assume that βi > 0 , since otherwise
node i would effectively be outside the financial system. Recall that wi is i’s initial net worth (before a
shock hits), and ci is the initial value of its outside assets. The shock Xi to node i takes values in [0, ci].
We shall assume that wi > 0, since otherwise i would already be insolvent. We shall also assume that
wi ≤ ci, since otherwise i could never default directly through losses in its own outside assets. Define
the ratio λi = ci/wi ≥ 1 to be the leverage of i’s outside assets. (This is not the same as i’s overall
leverage, which in our terminology is the ratio of i’s total assets to i’s net worth.)
3.1 A general bound on the probability of contagion
Proposition 1. Suppose that only node i receives a shock, so that Xj = 0 for all j 6= i. Suppose that
no nodes are in default before the shock. Fix a set of nodes D not containing i. The probability that the7Elliott, Golub, and Jackson (2013) have a similar measure which they call the level of integration. More broadly, Shin
(2012) discusses the reliance of banks on wholesale funding as a contributor to financial crises, and βi measures the degreeof this reliance in our setting.
8
shock causes all nodes in D to default is at most
P (Xi ≥ wi + (1/βi)∑j∈D
wj). (8)
Moreover, contagion from i to D is impossible if∑j∈D
wj/wi > βi(λi − 1). (9)
The condition in (9) states that contagion from i to D is impossible if the total net worth of the
nodes in D is sufficiently large relative to the net worth of i weighted by the exposure of the financial
system to node i, as measured by βi, and the vulnerability of i as measured by its leverage. A similar
interpretation applies to (8).
Before proceeding to the proof, we illustrate the impossibility condition in (9) through the network
in Figure 2(a). The central node is node i, meaning that the shock affects its outside assets, and the
remaining nodes comprise D. The relevant parameter values are βi = 2/7, λi = 15, and the net worths
are as indicated in the figure. The left side of (9) evaluates to 2 and the right side to 4, so the condition
is violated, and, indeed, we saw earlier that contagion is possible with a shock greater than 80. However,
a modification of the network that raises the sum of the net worths of the peripheral nodes above 40
makes contagion impossible. This holds, for example, if the outside liabilities of every peripheral node
are reduced by more than 5, or if the outside liabilities of a single peripheral node are reduced by more
than 20. This example also illustrates that (9) is tight in the sense that if the reverse strict inequality
holds, then contagion is possible in this example with a sufficiently large shock.
Proof of Proposition 1. Let D(x) ≡ D be the default set resulting from the shock vector X, whose
coordinates are all zero except for Xi. By assumption i causes other nodes to default, hence i itself must
default, that is, i ∈ D. To prove (8) it suffices to show that
βi(Xi − wi) ≥∑
j∈D−{i}
wj ≥∑j∈D
wj . (10)
The second inequality in (10) follows from the assumption that no nodes are in default before the shock
and the fact that we must have D ⊆ D − {i} for all nodes in D to default.
For the first inequality in (10), define the shortfall at node j to be the difference sj = pj − pj . From
(3) we see that the vector of shortfalls s satisfies
s = (sA− w + X)+ ∧ p.
By (4) we have sj > 0 for j ∈ D and sj = 0 otherwise. We use a subscript D as in sD or AD to restrict
a vector or matrix to the entries corresponding to nodes in the set D. Then the vector of shortfalls at
9
the nodes of D satisfies
sD ≤ sDAD − wD + XD, (11)
hence
XD − wD ≥ sD(ID −AD). (12)
The vector sD is strictly positive in every coordinate. From the definition of βj we also know that the
jth row sum of ID −AD is at least 1− βj . Hence,
sD(ID −AD) · 1D ≥∑j∈D
sj(1− βj) ≥ si(1− βi). (13)
From (11) it follows that the shortfall at node i is at least as large as the initial amount by which i
defaults, that is,
si ≥ Xi − wi > 0. (14)
From (12)–(14) we conclude that∑j∈D
(Xj − wj) = Xi − wi −∑
j∈D−{i}
wj ≥ si(1− βi) ≥ (Xi − wi)(1− βi). (15)
This establishes (10) and the first statement of the proposition. The second statement follows from
the first by recalling that the shock to the outside assets cannot exceed their value, that is, Xi ≤ ci.
Therefore by (8) the probability of contagion is zero if ci < wi + (1/βi)∑
j∈D wj . Dividing through by
wi we see that this is equivalent to the condition∑
j∈D wj/wi > βi(λi − 1). �
The preceding proposition relates the probability of contagion from a given node i to the net worth
of the defaulting nodes in D relative to i’s net worth. The bounds are completely general and do not
depend on the distribution of shocks or on the topology of the network. The critical parameters are βi,
the degree to which the triggering node is indebted to the financial sector, and λi, the degree of leverage
of i’s outside assets.
The Appendix gives estimates of these parameters for large European banks, based on data from
stress tests conducted by the European Banking Authority (2011). Among the 50 largest of these banks
the average of the λi is 24.9, the average of our estimated βi is 14.9%, and the average of the products
βi(λi − 1) is 3.2. Proposition 1 implies that contagion from a “typical” bank i cannot topple a set of
banks D if the net worth of the latter is more than 3.2 times the net worth of the former, unless there
are additional channels of contagion.8
8On average, commercial banks in the United States are leveraged only about half as much as European banks, andtheir values of βi are somewhat smaller (Federal Reserve Release H.8, Assets and Liabilities of Commercial Banks in theUnited States, 2012). This suggests that contagion is even less likely in the US financial sector than in Europe.
10
3.2 Contagion with proportional shocks
We can say a good deal more if we impose some structure on the distribution of shocks. The notion
that contagion from i to D is weak, described informally in Section 1, can now be made precise by the
condition
P (Xi ≥ wi + (1/βi)∑j∈D
wj) ≤ P (Xi > wi)∏j∈D
P (Xj > wj). (16)
The expression on the left bounds the probability that these nodes default solely through contagion from
i, while the expression on the right is the probability that the nodes in D default through independent
direct shocks. Contagion is weak if the latter probability is at least as large as the former. The assumption
of independent direct shocks is somewhat unrealistic: in practice one would expect the shocks to different
nodes to be positively associated. In this case, however, the probability of default from direct shocks is
even larger, hence weak contagion covers this situation as well.
Let us assume that the losses at a given node i scale with the size of the portfolio ci. Let us also
assume that the distribution of these relative losses is the same for all nodes, and independent among
nodes. Then there exists a distribution function H : [0, 1] → [0, 1] such that
F (x1, ..., xn) =∏
1≤i≤n
H(xi/ci). (17)
Beta distributions provide a flexible family with which to model the distribution of shocks as a fraction
of outside assets. We work with beta densities of form
hp,q(y) =yp−1(1− y)q−1
B(p, q), 0 ≤ y ≤ 1, p, q ≥ 1, (18)
where B(p, q) is a normalizing constant. The subset with p = 1 and q > 1 has a decreasing density and
seems the most realistic, but (18) is general enough to allow a mode anywhere in the unit interval. The
case q = 1, p > 1 has an increasing density and could be considered “heavy-tailed” in the sense that it
assigns greater probability to greater losses, with losses capped at 100 percent of outside assets.9
Theorem 1. Assume the shocks are i.i.d. beta distributed as in (18) and that the net worth of every
node is initially nonnegative. Let D be a nonempty subset of nodes and let i /∈ D. Contagion from i to
D is impossible if ∑j∈D
wj > wiβi(λi − 1) (19)
and it is weak if ∑j∈D
wj ≥ wiβi
∑j∈D
(λi − 1)/λj . (20)
9Bank capital requirements under Basel II and III standards rely on a family of loss distributions derived from aGaussian copula model. As noted by Tasche (2008) and others, these distributions can be closely approximated by betadistributions.
11
As noted after Proposition 1, the condition in (19) states that contagion from i to D is impossible if
the total net worth of the nodes in D is sufficiently large relative to the net worth of i weighted by the
exposure of the financial system to node i and the vulnerability of i as measured by its leverage. The
condition in (20) compares the total net worth of D relative to that of i with the leverage of i relative
to that of the nodes in D. With other parameters held constant, increasing the relative net worth of
D makes contagion weaker in the sense that it strengthens the inequality; increasing the leverage of i
relative to that of nodes in D has the opposite effect. Importantly, the two effects are mediated by βi,
which measures the exposure of the financial system to node i — a lower βi makes D less vulnerable
to i and makes D less sensitive to the degree of leverage at i. Thus, (20) captures the effects of equity
levels, leverage ratios, and the degree of reliance on interbank lending on the risk of contagion.
By recalling that λj = cj/wj we can rewrite (20) in the equivalent form∑j∈D
cjλ−1j /
∑j∈D
λ−1j ≥ ciβi(1− λ−1
i ). (21)
Written this way, the condition states that contagion from i to D is weak if the average size of the nodes
in D weighted by their inverse leverage ratios (their capital ratios) is sufficiently large relative to i; on
the right side of the inequality, ciβi measures the financial system’s exposure to node i’s outside assets,
and the factor (1− λ−1i ) is greater when node i is more highly leveraged. Inequality (21) is thus harder
to satisfy, and D more vulnerable to contagion from i, if the large nodes in D are more highly leveraged,
if node i draws more of its funding from the financial system, or if node i is more highly leveraged.
Through (21), a key implication of Theorem 1 is that without some heterogeneity, contagion will be
weak irrespective of the structure of the interbank network:
Corollary 1. Assume that all nodes hold the same amount of outside assets ck ≡ c. Under the assump-
tions of Theorem 1, contagion is weak from any node to any other set of nodes.
Proof. This follows from the fact that βi(1 − λ−1i ) < 1; hence, if ck ≡ c, inequality (21) holds for all i
and D. �
In the Appendix we apply our framework to the 50 largest banks in the stress test data from the
European Banking Authority. It turns out that contagion is weak in a wide variety of scenarios. In
particular, we analyze the scenario in which one of the five largest European banks (as measured by
assets) topples two other banks in the top 50. We find that the probability of such an event is less than
the probability of direct default unless the two toppled banks are near the bottom of the list of 50.
In the example of Figure 2(a), with node i the central node, the left side of (20) evaluates to 20, and
the right side evaluates to 16 because βi = 2/7, λi = 15, and each of the peripheral nodes has λj = 10.
Thus, contagion is weak.
12
Proof of Theorem 1. Proposition 1 implies that contagion is weak from i to D if
P (Xi ≥ wi + (1/βi)∑j∈D
wj) ≤ P (Xi > wi)∏j∈D
P (Xj > wj). (22)
On the one hand this certainly holds if wi + (1/βi)∑
j∈D wj > ci, for then contagion is impossible. In
this case we obtain, as in (9), ∑j∈D
wj/wi > βi(λi − 1). (23)
Suppose on the other hand that (wi + (1/βi)∑
j∈D wj) ≤ ci. By assumption the relative shocks Xk/ck
are independent and beta distributed as in (18). In the uniform case p = q = 1, (22) is equivalent to
[1− (wi/ci + (1/βici)∑j∈D
wj)] ≤ (1− wi/ci)∏j∈D
(1− wj/cj), (24)
We claim that (24) implies (22) for the full family of beta distributions in (19). To see why, first observe
that the cumulative distribution Hp,q of hp,q satisfies
1−Hp,q(y) = Hq,p(1− y).
Hence (22) holds if
Hq,p(1− wi/ci − (1/βici)∑j∈D
wj) ≤ Hq,p(1− wi/ci)∏j∈D
Hq,p(1− wj/cj). (25)
But (25) follows from (24) because beta distributions with p, q ≥ 1 have the submultiplicative property
Hq,p(xy) ≤ Hq,p(x)Hq,p(y), x, y ∈ [0, 1].
(See Proposition 4.1.2 of Wirch 1999; the application there has q ≤ 1, but the proof remains valid for
q ≥ 1. The inequality can also be derived from Corollary 1 of Ramos Romero and Sordo Diaz 2001.) It
therefore suffices to establish (25), which is equivalent to
(1/βici)∑j∈D
wj ≥ (1− wi/ci)(1−∏j∈D
(1− wj/cj)). (26)
Given any real numbers θj ∈ [0, 1] we have the inequality∏j
(1− θj) ≥ 1−∑
j
θj . (27)
Hence a sufficient condition for (26) to hold is that
(1/βici)∑j∈D
wj ≥ (1− wi/ci)(∑j∈D
wj/cj). (28)
13
After rearranging terms and using the fact that λk = ck/wk for all k, we obtain (20). This concludes
the proof of Theorem 1. �
From the argument following (25), it is evident that the same result holds if the shocks to each node
j are distributed with parameters pj,qj in (18) with pi ≤ minj∈D pj and qi ≥ maxj∈D qj .
As a further illustration of Theorem 1, suppose the nodes in D are numbered 2, . . . ,m and suppose
node i = 1 receives a shock. Further suppose the outside assets are ordered c1 ≥ c2 ≥ · · · ≥ cm; because
shocks are proportional to outside assets, the assumption that the node with the largest ck receives the
shock maximizes the chances of contagion to the other nodes.
Corollary 2. If c1 ≥ c2 ≥ · · · ≥ cm, then contagion from node 1 to nodes 2, . . . ,m is weak if c2 ≥
β1(c1−w1) and cj ≥ (cj−1−wj−1), j = 2, . . . ,m. Contagion is impossible if, in addition, c2−cm+wm >
β1(c1 − w1).
This is a direct consequence of (24) and (26), hence we omit the details and comment on the in-
terpretation. The lower bounds on the cj ensure that the potential spillovers from other nodes cannot
push the full set of nodes into default regardless of the network topology. Viewing the conditions in the
corollary as lower bounds on the wj suggests minimum capital requirements to ensure the resilience of
the system, based on the relative sizes of banks.
Of course these results do not say that the network structure has no effect on the probability of
contagion; indeed there is a considerable literature showing that it does (see among others Haldane
and May 2011, Gai and Kapadia 2011, Georg, 2011). Rather it shows that in quite a few situations
the probability of contagion will be lower than the probability of direct default, absent some channel of
contagion beyond spillovers through payment obligations. We have already mentioned bankruptcy costs,
fire sales, and mark-to-market losses as amplifying mechanisms. The models of Demange (2012) and
Acemoglu, Ozdaglar, and Tahbaz-Salehi (2013) generate greater contagion by making debts to financial
institutions subordinate to other payment obligations. With priority given to outside payments, shocks
produce greater losses within the network. In practice, bank debt and bank deposits are owned by both
financial institutions and non-financial firms and individuals, so characterizing seniority based on the
type of lender is problematic.
3.3 Contagion with truncated shocks
In this section we shall show that the preceding results are not an artifact of the beta distribution:
similar bounds hold for a variety of shock distributions. Under the beta distribution the probability is
zero that a node loses all of its outside assets. One could easily imagine, however, that the probability
of this event is positive. This situation can be modeled as follows. Let Xoi ≥ 0 be a primary shock
14
(potentially unbounded in size) and let Xi = ci ∧ Xoi be the resulting loss to i’s outside assets. For
example Xoi might represent a loss of income from an employment shock that completely wipes out i’s
outside assets. Assume that the primary shocks have a joint distribution function of form
F o(xo1, ..., x
on) =
∏1≤i≤n
Ho(xoi /ci), (29)
where Ho is a distribution function on the nonnegative real line. In other words we assume that the
shocks are i.i.d. and that a given shock xoi affects every dollar of outside assets ci equally. (We shall
consider a case of dependent shocks after the next result.)
In general a random variable with distribution function G and density g is said to have an increasing
failure rate (IFR) distribution if g(x)/(1 − G(x)) is an increasing function of x. Examples of IFR
distributions include all normal, exponential, and uniform distributions and, more generally, all log-
concave distributions. Observe that truncating the shock can put mass at ci and thus assign positive
probability to a total loss of outside assets.
Theorem 2. Assume the primary shocks are i.i.d. and IFR-distributed, and that the net worth of every
node is initially nonnegative. Let D be a nonempty subset of nodes and let i /∈ D. Contagion from i to
D is impossible if ∑j∈D
wj > wiβi(λi − 1)
and it is weak if ∑j∈D
wj > wiβi
∑j∈D
λi/λj . (30)
Corollary 3. Assume that all nodes hold the same amount of outside assets ci ≡ c. Under the assump-
tions of Theorem 2, contagion is weak from any node to any other set of nodes.
This is immediate upon rewriting (30) as∑j∈D
cjλ−1j /
∑j∈D
λ−1j ≥ βici.
Proof of Theorem 2 . Through relabeling, we can assume that the source node for contagion is i = 1
and that the infected nodes are D = {2, 3, . . . ,m}. By Proposition 1 we know that contagion is weak
from 1 to D if
P (X1 > w1 + (1/β1)∑
2≤j≤m
wj) ≤ P (X1 > w1)P (X2 > w2) · · ·P (Xm > wm). (31)
15
Since X1 = c1 ∧ Xo1 , the left-hand side is zero when w1 + (1/β1)
∑2≤j≤m wj > c1. Thus contagion is
impossible if ∑2≤j≤m
wj/w1 > β1(λ1 − 1). (32)
Let us therefore assume that w1 + (1/β1)∑
2≤j≤m wj ≤ c1. Define the random variables Yi = Xoi /ci.
Weak contagion from 1 to D holds if
P (Y1 > w1/c1 + (1/β1c1)∑
2≤j≤m
wj) ≤ P (Y1 > w1/c1)P (Y2 > w2/c2) · · ·P (Ym > wm/cm)
= P (Y1 > w1/c1)P (Y1 > w2/c2) · · ·P (Y1 > wm/cm). (33)
where the latter follows from the assumption that the Yi are i.i.d. By assumption Y1 is IFR, hence
P (Y1 > s + t|Y1 > s) ≤ P (Y1 > t) for all s, t ≥ 0. (See for example Barlow and Proschan 1975, p.159.)
It follows that
P (Y1 >∑
1≤k≤m
wk/ck)
= P (Y1 > w1/c1)P (Y1 > w1/c1 + w2/c2|Y1 > w1/c1)
· · ·P (Y1 > w1/c1 + w2/c2 + · · ·+ wm/cm|Y1 > w1/c1 + w2/c2 + · · ·+ wm−1/cm−1)
≤ P (Y1 > w1/c1)P (Y1 > w2/c2) · · ·P (Y1 > wm/cm)
Together with (33) this shows that contagion from 1 to D is weak provided that
P (Y1 ≥ w1/c1 + (1/β1c1)∑
2≤j≤m
wj) ≤ P (Y1 >∑
1≤k≤m
wk/ck). (34)
This clearly holds if
w1/c1 + (1/β1c1)∑
2≤j≤m
wj ≥∑
1≤k≤m
wk/ck, (35)
which is equivalent to
(1/β1c1)∑
2≤j≤m
wj ≥∑
2≤j≤m
wj/cj =∑
2≤j≤m
λ−1j . (36)
Since c1 = λ1w1, we can re-write (36) as∑2≤j≤m
wj/w1 ≥ β1λ1
∑2≤j≤m
λ−1j . (37)
We have therefore shown that if contagion from 1 to D = {2, 3, . . . ,m} is possible at all, then (37) is a
sufficient condition for weak contagion. �
From (37), we see that a simple sufficient condition for weak contagion is cj ≥ β1c1, j = 2, . . . ,m,
and the condition∑m
j=2 wj > β1(c1 − w1) makes contagion impossible.
16
The assumption of independent shocks to different nodes is conservative. If we assume that direct
shocks to different nodes are positively dependent, as one would expect in practice, the bounds in
Theorems 1 and 2 will be lower and the relative likelihood of default through contagion even smaller.
A particularly simple case arises when the primary shocks Xi are independent with a negative expo-
nential distribution of form
f(xi) = µe−µxi , xi ≥ 0. (38)
If we further assume that c1 = · · · = cm ≡ c and β1 = 1, then the two probabilities compared in (33) are
equal, both evaluating to exp(−µ∑
j∈D wj/c). In this sense, the exponential distribution is a borderline
case in which the probability of a set of defaults from a single shock is roughly equal to the probability
from multiple independent shocks. We say “roughly” because the left side of (33) is an upper bound on
the probability of default through contagion, and in practice the βi are substantially smaller than 1. In
the example of Figure 2(a), we have seen that contagion from the central node requires a shock greater
than 80, which has probability exp(−80µ) under an exponential distribution. For direct defaults, it
suffices to have shocks greater than 5 at the peripheral nodes and a shock greater than 10 at the central
node, which has probability exp(−30µ) given i.i.d. exponential shocks.
If the primary shocks have a Pareto-like tail, meaning that
P (Xi > x) ∼ ax−µ (39)
for some positive constants a and µ (or, more generally, a regularly varying tail), then the probability
that a single shock will exceed∑
j∈D wj/c will be greater than the probability that the nodes in D
default through multiple independent shocks, at least at large levels of the wj . However, introducing
some dependence can offset this effect, as we now illustrate. To focus on the issue at hand, we take c = 1
and βi = 1.
To consider a specific and relatively simple case, let Y1, . . . , Ym be independent random variables,
each distributed as tν , the Student t distribution with ν > 2 degrees of freedom. Let Y1, . . . , Ym have
a standard multivariate Student t distribution with tν marginals.10 The Yj are uncorrelated but not
independent. To make the shocks positive, set Xj = Y 2j and Xj = Y 2
j . Each Xj and Xj has a Pareto-like
tail that decays with a power of ν/2.
Proposition 2. With independent shocks Xj,
P (Xi >m∑
j=1
wj) ≥m∏
j=1
P (Xj > wj)
10More explicitly, (Y1, ..., Ym) has the distribution of (Z1, ..., Zm)/p
χ2ν/ν, where the Zi are independent standard normal
random variables and χ2v has a chi-square distribution with v degrees of freedom and is independent of the Zi.
17
for all sufficiently large wj, j = 1, . . . ,m. With dependent shocks
P (Xi >m∑
j=1
wj) ≤ P (Xj > wj , j = 1, . . . ,m)
for all wj ≥ 0, j = 1, . . . ,m.
Proof of Proposition 2 . The first statement follows from applying (39) to both sides of the inequality.
The second statement is an application of Bound II for the F distribution on p.1196 of Marshall and
Olkin (1974). �
Thus, even with heavy-tailed shocks, we may find that default of a set of nodes through contagion
from a single shock is less likely than default through direct shocks to individual nodes if the shocks are
dependent.
4 Amplification of losses due to network effects
The preceding analysis dealt with the impact of default by a single node (the source) on another set of
nodes (the target). Here we shall examine the impact of shocks on the entire system, including multiple
and simultaneous defaults. To carry out such an analysis, we need to have a measure of the total systemic
impact of a shock. There appears to be no commonly accepted measure of systemic impact in the prior
literature. Eisenberg and Noe (2001) suggest that it is the number of waves of default that a given shock
induces in the network. Other authors have suggested that the systemic impact should be measured
by the aggregate loss of bank capital; see for example Cont, Moussa, and Santos (2010). Still others
have proposed the total loss in value of only those nodes external to the financial sector, i.e. firms and
households.
Here we shall take the systemic impact of a shock to be the total loss in value summed over all nodes,
including nodes corresponding to financial entities as well as those representing firms and households.
This measure is easily stated in terms of the model variables. Given a shock realization x, the total
reduction in asset values is
∑i xi + S(x) where S(x) =
∑i(pi − pi(x)). (40)
The term |x| =∑
i xi is the direct loss in value from reductions in payments by the external sector.
The term S(x) is the indirect loss in value from reductions in payments by the nodes to other nodes and
to the external sector. An overall measure of the riskiness of the system is the expected loss in value
L =∫
(|x|+ S(x))dF (x). (41)
The question we wish to examine is what proportion of these losses can be attributed to connections
between institutions as opposed to characteristics of individual banks. To analyze this issue let x be
18
a shock and let D = D(x) be the set of nodes that defaults given x. Under our assumptions this set
is unique because the clearing vector is unique. To avoid notational clutter we shall suppress x in the
ensuing discussion.
As in the proof of Propostion 1, define the shortfall in payments at node i to be si = pi − pi, where
p is the clearing vector. By definition of D,
si > 0 for all i ∈ Dsi = 0 for all i /∈ D
(42)
Also as in the proof of Proposition 1, let AD be the |D| × |D| matrix obtained by restricting the relative
liabilities matrix A to D, and let ID be the |D| × |D| identity matrix. Similarly let sD be the vector of
shortfalls si corresponding to the nodes in D, let wD be the corresponding net worth vector defined in
(1), and let xD be the corresponding vector of shocks. The clearing condition (3) implies the following
shortfall equation, provided no node is entirely wiped out — that is, provided si < pi, for all i:
sDAD − (wD − xD) = sD. (43)
Allowing the possibility that some si = pi, the left side is an upper bound on the right side. Recall that
AD is substochastic, that is, every row sum is at most unity. Moreover, by assumption, there exists a
chain of obligations from any given node k to a node having strictly positive obligations to the external
sector. It follows that limk→∞AkD = 0D, hence ID −AD is invertible and
[ID −AD]−1 = ID + AD + A2D + .... (44)
From (43) and (44) we conclude that
sD = (xD − wD)[ID + AD + A2D + ...]. (45)
Given a shock x with resulting default set D = D(x), define the vector u(x) ∈ Rn+ such that
uD(x) = [ID + AD + A2D + ...] · 1D , ui(x) = 0 for all i 6∈ D. (46)
Combining (40), (45), and (46) shows that total losses given a shock x can be written in the form
L(x) =∑
i
(xi ∧ wi) +∑
i
(xi − wi)ui(x). (47)
The first term represents the direct losses to equity at each node and the second term represents the
total shortfall in payments summed over all of the nodes. The right side becomes an upper bound on
L(x) if si = pi for some i ∈ D(x).
We call the coefficient ui = ui(x) the depth of node i in D = D(x). The rationale for this terminology
is as follows. Consider a Markov chain on D with transition matrix AD. For each i ∈ D, ui is the expected
19
number of periods before exiting D, starting from node i.11 Expression (46) shows that the node depths
measure the amplification of losses due to interconnections among nodes in the default set.
We remark that the concept of node depth is dual to the notion of eigenvector centrality in the
networks literature (see for example Newman 2010). To see the connection let us restart the Markov
chain uniformly in D whenever it exits D. This modified chain has an ergodic distribution proportional
to 1D · [ID + AD + A2D + ...], and its ergodic distribution measures the centrality of the nodes in D. It
follows that node depth with respect to AD corresponds to centrality with respect to the transpose of
AD.
Although they are related algebraically, the two concepts are quite different. To see why let us return
to the example of Figure 2(b). Suppose that node 1 (the central node) suffers a shock x1 > 80. This
causes all nodes to default, that is, the default set is D′ = {1, 2, 3, 4, 5}. Consider any node j > 1. In
the Markov chain described above the expected waiting time to exit the set D′, starting from node j, is
given by the recursion uj = 1 + βjuj , which implies
uj = 1/(1− βj) = 1 + y/55. (48)
From node 1 the expected waiting time satisfies the recursion
u1 = 1(100/140) + (1 + uj)(40/140). (49)
Hence
u1 = 9/7 + 2y/385. (50)
Comparing (48) and (50) we find that node 1 is deeper than the other nodes (u1 > uj) for 0 ≤ y < 22
and shallower than the other nodes for y > 22. In contrast, node 1 has lower eigenvector centrality than
the other nodes for all y ≥ 0 because it cannot be reached directly from any other node.
The magnitude of the node depths in a default set can be bounded as follows. In the social networks
literature a set D is said to be α-cohesive if every node in D has at least α of its obligations to other
nodes in D, that is,∑
j∈D aij ≥ α for every i ∈ D (Morris, 2000). The cohesiveness of D is the maximum
such α, which we shall denote by αD. From (46) it follows that
∀i ∈ D, ui ≥ 1/(1− αD). (51)
Thus the more cohesive the default set, the greater the depth of the nodes in the default set and the
greater the amplification of the associated shock.
Similarly we can bound the node depths from above. Recall that βi is the proportion of i’s obligations
to other nodes in the financial system. Letting βD = max{βi : i ∈ D} we obtain the upper bound,11Liu and Staum (2012) show that the node depths can be used to characterize the gradient of the clearing vector p(x)
with respect to the asset values.
20
assuming βD < 1,
∀i ∈ D, ui ≤ 1/(1− βD) . (52)
The bounds in (51) and (52) depend on the default set D, which depends on the shock x. A uniform
upper bound is given by
∀i, ui ≤ 1/(1− β+) where β+ = maxβi, (53)
assuming β+ < 1.
We are now in a position to compare the expected systemic losses in a given network of interconnec-
tions, and the expected systemic losses without such interconnections. As before, fix a set of n nodes
N = {1, 2, ..., n}, a vector of outside assets c = (c1, c2, ..., cn) ∈ Rn+ and a vector of outside liabilities
b = (b1, b2, ..., bn) ∈ Rn+. Assume the net worth wi of node i is nonnegative before a shock is realized.
Interconnections are determined by the n× n liabilities matrix P = (pij).
Let us compare this situation with the following: eliminate all connections between nodes, that is,
let P o be the n × n matrix of zeroes. Each node i still has outside assets ci and outside liabilities bi.
To keep their net worths unchanged, we introduce “fictitious” outside assets and liabilities to balance
the books. In particular if ci − bi < wi we give i a new class of outside assets in the amount c′i =
wi− (ci− bi). If ci− bi > wi we give i a new class of outside liabilities in the amount b′i = wi− (ci− bi).
We shall assume that these new assets are completely safe (they are not subject to shocks), and that
the new liabilities have the same priority as all other liabilities.
Let F (x1, ..., xn) be a joint shock distribution that is homogeneous in assets, that is, F (x1, ..., xn) =
G(x1/c1, ..., xn/cn) where G is a symmetric c.d.f. (Unlike in the preceding results on contagion we do
not assume that the shocks are independent.) We say that F is IFR if its marginal distributions are
IFR; this is equivalent to saying that the marginals of G are IFR. Given F , let L be the expected total
losses in the original network and let Lo be the expected total losses when the connections are removed
as described above.
Theorem 3. Let N(b, c, w, P ) be a financial system and let No be the analogous system with all the
connections removed. Assume that the shock distribution is homogeneous in assets and IFR. Let β+ =
maxi βi < 1 and let δi = P (Xi ≥ wi). The ratio of expected losses in the original network to the expected
losses in No is at mostL
Lo≤ 1 +
∑δici
(1− β+)∑
ci. (54)
Proof. By assumption the marginals of F are IFR distributed. A general property of IFR distributions
is that “new is better than used in expectation,” that is,
∀i ∀wi ≥ 0, E[Xi − wi|Xi ≥ wi] ≤ E[Xi] (55)
21
(Barlow and Proschan 1975, p.159). It follows that
∀i ∀wi ≥ 0, E[(Xi − wi)+] ≤ P (Xi ≥ wi)E[Xi] = δiE[Xi]. (56)
By (47) we know that the total expected losses L can be bounded as
L ≤∑
i
E[Xi ∧ wi] + E[∑
i
(Xi − wi)ui(X)]. (57)
From (53) we know that ui ≤ 1/(1 − β+) for all i; furthermore we clearly have Xi − wi ≤ (Xi − wi)+
for all i. Therefore
L ≤∑
i
E[Xi ∧ wi] + (1− β+)−1∑
i
E[(Xi − wi)+]. (58)
From this and (56) it follows that
L ≤∑
i
E[Xi ∧ wi] + (1− β+)−1∑
δiE[Xi]
≤∑
i
E[Xi] + (1− β+)−1∑
δiE[Xi]. (59)
When the network connections are excised, the expected loss is simply the expected sum of the shocks,
that is, Lo=∑
i E[Xi]. By the assumption of homogeneity in assets we know that E[Xi] ∝ ci for all i.
We conclude from this and (59) that
L/Lo ≤ 1 +∑
δici
(1− β+)∑
ci.
This completes the proof of Theorem 3. �
Theorem 3 shows that the increase in losses due to network interconnections will be very small unless
β+ is close to 1 or the default rates of some banks (weighted by their outside asset base) is large.
Moreover this statement holds even when the shocks are dependent, say due to common exposures, and
it holds independently of the network structure.
Some might argue that write-downs of purely financial obligations should not be counted as part of
the systemic loss. This is tantamount to truncating node depth at 1 and thus leads to an even smaller
upper bound in Theorem 3.
To illustrate the result concretely, consider the EBA data. In this case β+ = max{βi} = 0.43 where
the maximum occurs for Dexia bank. Regulatory restrictions on assets imply that any given bank is
unlikely to fail due to direct default; indeed current regulatory standards seek to make the direct default
probability of any individual bank smaller than 0.1% per year. Let us make the more conservative
assumption that the probability of default is at most 1%, that is, max{δi} < 0.01. If this standard is
met, then no matter what the interconnections might be among the banks in the data set, we conclude
22
from Theorem 3 that the additional expected loss attributable to the network is at most .01/(1− .43),
which is about 1.7%. In other words the actual network of connections cannot increase the expected
losses by very much under the assumptions of the theorem.
5 Bankruptcy costs
In this section and the next we enrich the basic framework by incorporating additional mechanisms
through which losses propagate from one node to another. We begin by adding bankruptcy costs. The
equilibrium condition (3) implicitly assumes that if node i’s assets fall short of its liabilities by 1 unit,
then the total claims on node i are simply marked down by 1 unit below the face value of pi. In practice,
the insolvency of node i is likely to produce deadweight losses that have a knock-on effect on the shortfall
at node i and at other nodes.
Cifuentes, Ferrucci, and Shin (2005), Battiston et al. (2012), Cont, Moussa, and Santos (2010),
and Rogers and Veraart (2012) all use some form of liquidation cost or recovery rate at default in their
analyses. Cifuentes, Ferrucci, and Shin (2005) distinguish between liquid and illiquid assets and introduce
an external demand function to determine a recovery rate on illiquid assets. Elliott, Golub, and Jackson
(2013) attach a fixed cost to bankruptcy. Elsinger, Lehar, and Summer (2006) present simulation results
illustrating the effect of bankruptcy costs but without an explicit model. The mechanism we use appears
to be the simplest and the closest to the original Eisenberg-Noe setting, which facilitates the analysis of
its impact on contagion.
5.1 Shortfalls with bankruptcy costs
In the absence of bankruptcy costs, when a node fails its remaining assets are simply divided among
its creditors. To capture costs of bankruptcy that go beyond the immediate reduction in payments, we
introduce a multiplier γ ≥ 0 and suppose that upon a node’s failure its assets are further reduced by
γ
pi − (ci +∑j 6=i
pjaji − xi)
, (60)
up to a maximum reduction at which the assets are entirely wiped out. This approach is analytically
tractable and captures the fact that large shortfalls are considerably more costly than small shortfalls,
where the firm nearly escapes bankruptcy. The term in square brackets is the difference between node
i’s obligations pi and its remaining assets. This difference measures the severity of the failure, and the
factor γ multiplies the severity to generate the knock-on effect of bankruptcy above and beyond the
immediate cost to node i’s creditors. We can think of the expression in (60) as an amount of value
destroyed or paid out to a fictitious bankruptcy node upon the failure of node i.
23
The resulting condition for a payment vector replaces (3) with
pi = pi ∧
(1 + γ)(ci +∑j 6=i
pjaji − xi)− γpi
+
. (61)
Written in terms of shortfalls si = pi − pi, this becomes
si = (1 + γ)[∑j 6=i
sjaji − wi + xi]+ ∧ pi. (62)
Here we see explicitly how the bankruptcy cost factor γ magnifies the shortfalls.
Let Φsγ denote the mapping from the vector s on the right side of (62) to the vector s on the left
side, and let Φpγ similarly denote the mapping of p defined by (61). We have the following:
Proposition 3. (a) For any γ ≥ 0, the mapping Φsγ is monotone increasing, bounded, and continuous
on Rn+, and the mapping Φp
γ is monotone decreasing, bounded, and continuous. It follows that Φsγ has a
least fixed point s and Φpγ has a greatest fixed point p = p − s. (b) For any s, Φs
γ(s) is increasing in γ,
and for any p, Φpγ(p) is decreasing in γ. Consequently, the set of default nodes under the minimal s and
maximal p is increasing in γ. The fixed point s (and p) is unique if (1 + γ)A has spectral radius less
than 1.
Proof of Proposition 3. Part (a) follows from the argument in Theorem 1 of Eisenberg and Noe (2001).
For part (b), write vi = ci + (pA)i − xi and observe that
Φpγ(p)i =
{vi − γ(pi − vi), vi < pi;pi otherwise,
from which the monotonicity in γ follows. The maximal fixed-point is the limit of iterations of Φpγ
starting from p by the argument in Section 3 of Eisenberg and Noe (2001). If γ1 ≤ γ2, then the iterates
of Φpγ1
are greater than those of Φpγ2
, so their maximal fixed-points are ordered the same way. But then
the set of nodes i for which pi < pi at the maximal fixed-point for γ1 must be contained within that for
γ2. The same argument works for Φsγ . Uniqueness follows as in the case without bankruptcy costs. �
This result confirms that bankruptcy costs expand the set of defaults (i.e., increase contagion) result-
ing from a given shock realization x while otherwise leaving the basic structure of the model unchanged.
In what follows, we examine the amplifying effect of bankruptcy costs conditional on a default set D,
and we then compare costs with and without network effects. The factor of 1 + γ already points to the
amplifying effect of bankruptcy costs, but we can take the analysis further.12
Suppose, for simplicity, that the maximum shortfall of pi in (62) is not binding on any of the nodes
in the default set. In other words, the shocks are large enough to generate defaults, but not so large as12A corresponding comparison is possible using partial recoveries at default, as in Rogers and Veraart (2012). This leads
to qualitatively similar results, provided claims on other banks are kept to a realistic fraction of a bank’s total assets.
24
to entirely wipe out asset value at any node. In this case, we have
sD = (1 + γ)[sDAD − wD + xD].
If we further assume that ID − (1 + γ)AD is invertible, then
sD = (1 + γ)(xD − wD)[ID − (1 + γ)AD]−1
and the systemic shortfall is
S(x) = sD · uD(γ) = (1 + γ)(xD − wD) · uD(γ), (63)
where the modified node depth vector uD(γ) is given by [ID − (1 + γ)AD]−1 · 1D. If (1 + γ)AD has
spectral radius less than 1, then
uD(γ) = [ID + (1 + γ)AD + (1 + γ)2A2D + · · · ] · 1D.
The representation in (63) reveals two effects from introducing bankruptcy costs. The first is an imme-
diate or local impact of multiplying xi−wi by 1 + γ; every element of sD is positive, so the outer factor
of 1 + γ increases the total shortfall. The second and more important effect is through the increased
node depth. In particular, letting αD denote the cohesiveness of D as before, we can now lower-bound
the depth of each node by 1/[1 − (1 + γ)αD].13 This makes explicit how bankruptcy costs deepen the
losses at defaulted nodes and increase total losses to the system. By the argument in Theorem 3, we get
the following comparison of losses with and without interconnections:
Corollary 4. In the setting of Theorem 3, if we introduce bankruptcy costs satisfying (1 + γ)β+ < 1,
thenL
Lo≤ 1 +
∑δici
[1− (1 + γ)β+]∑
ci.
In the EBA data in the appendix, we have β+ = 0.43. If we set γ at the rather large value of 0.5 and
continue to assume δi ≤ 0.01, the corollary gives an upper bound of 1.042. In other words, even with
large bankruptcy costs, the additional expected loss attributable to the network is at most 4.2%.
5.2 Example
We saw previously that in the example of Figure 2(a) we need a shock greater than 80 to the outside
assets of the central node in order to have contagion to all other nodes. Under a beta distribution13If the spectral radius of (1 + γ)A is less than 1, then (1 + γ)αD < 1.
25
with parameters p = 1 and q ≥ 1, this has probability (1 − 80/150)q = (7/15)q. If we assume i.i.d.
proportional shocks to the outside assets of all nodes, then all nodes default directly with probability
[(1− w1/c1) · · · (1− w5/c5)]q = [(14/15) · 0.94]q ≈ 0.61q > (7/15)q.
Thus contagion is weak.
Now introduce a bankruptcy cost factor of γ = 0.5 and consider a shock of 57 or larger to the
central node. The shock creates a shortfall of at least 47 at the central node that gets magnified by
50% to a shortfall of at least 70.5 after bankruptcy costs. The central node’s total liabilities are 140,
so the result is that each peripheral node receives less than half of what it is due from the central
node, and this is sufficient to push every peripheral node into default. Thus, with bankruptcy costs, the
probability of contagion is at least (1− (57/150))q = 0.62q, which is now greater than the probability of
direct defaults through independent shocks. A similar comparison holds with truncated exponentially
distributed shocks.
This example illustrates how bankruptcy costs increase the probability of contagion. However, it is
noteworthy that γ needs to be quite large to overcome the effect of weak contagion.
6 Confidence, credit quality and mark-to-market losses
In the previous section, we demonstrated how bankruptcy costs can amplify losses. In fact, a borrower’s
deteriorating credit quality can create mark-to-market losses for a lender well before the point of default.
Indeed, by some estimates, these types of losses substantially exceeded losses from outright default during
2007-2009. We introduce a mechanism for adding this feature to a network model and show that it too
magnifies contagion. See Harris, Herz, and Nissim (2012) for a broad discussion of accounting practices
as sources of systemic risk.
The mechanism we introduce is illustrated in Figure 3, which shows how the value of node i’s total
liabilities changes with the level z of node i’s assets. The figure shows the special case of a piecewise
linear relationship. More generally, let r(z) be the reduced value of liabilities at a node as a function of
asset level z, where r(z) is increasing, continuous, and
0 ≤ r(z) ≤ pi, for z < (1 + k)pi;
r(z) = pi, for z ≥ (1 + k)pi.
Let R(z1, . . . , zn) = (r(z1), . . . , r(zn)). Given a shock x = (x1, . . . , xn), the clearing vector p(z) solves
p(x) = R(c + p(x)A + w − x).
Our conditions on r ensure the existence of such a fixed point by an argument similar to the one in
Section 2.2.
26
ip
ip
[1 (1 )] ik p zη η− + +
ipk)1( +
( )( ) 1ik p zγ η γ− + + +
Assets z
Value of Liabilities ( )r z
Figure 3: Liability value as a function of asset level in the presence of bankruptcy costs and creditquality.
The effect of credit quality deterioration begins at a much higher asset level of z = (1 + k)pi than
does default. Think of k as measuring a capital cushion: node i’s credit quality is impaired once its
net worth (the difference between its assets and liabilities) falls below the cushion. At this point, the
value of i’s liabilities begins to decrease, reflecting the mark-to-market impact of i’s deteriorating credit
quality.
In the example of Figure 2(a), suppose the central node is at its minimum capital cushion before
experiencing a shock; in other words (1 + k)pi = 150, which implies that k = 1/14. A shock of 5 to the
central node’s outside assets reduces the total value of its liabilities by 5η, so each peripheral node incurs
a mark-to-market loss of 5η/14. In contrast, in the original model with η = 0, the peripheral nodes do
not experience a loss unless the shock to the central node exceeds 10.
As this example illustrates, mark-to-market losses from credit deterioration and bankruptcy costs
operate in qualitatively different ways, even though both increase overall losses. Bankruptcy costs
amplify the propagation of shortfall. In contrast, accounting for credit quality effectively increases the
linkages between nodes. It does so by propagating losses at higher levels of asset values, rather than
by amplifying the losses as they are propagated. Shocks can also be propagated through prices due to
common exposures or fire sales that go beyond the network of direct obligations, as in Allen and Gale
(2000), Cifuentes, Ferrucci, and Shin (2005), and Caccioli et al. (2012).
7 Concluding Remarks
In this paper we have argued that it is relatively difficult to generate contagion solely through spillover
losses in a network of payment obligations. For contagion to occur, a shock to one node must lead to
losses at another set of nodes sufficiently large to wipe out their initial equity (or net worth). This means
that either the initial shock must be large (and therefore improbable) relative to the net worth of the
infecting node, or that the net worth of the infected nodes must be small. More generally, the probability
27
of contagion depends on the comparative net worths of the various nodes, their level of leverage, and
the extent to which the infecting node has obligations to the rest of the financial sector. As Theorems
1 and 2 show, one can compare the probability of contagion and direct default under a wide range of
shock distributions without knowing the topological details of the network.
The network structure matters more for the amplification effect, in which losses among defaulting
nodes multiply because of their obligations to one another. The degree of amplification is captured by
the concept of node depth, which is the expected number of periods it takes to exit from the default set
from a given starting point. This clearly depends on the specific structure of the interbank obligations,
and is dual to the concept of eigenvector centrality in the networks literature. As Theorem 3 shows one
can place an upper bound on the amplification effect based on banks’ maximum connectedness with the
rest of the financial system (the parameter β+) without knowing any details of the connections within
the system.
The network structure takes on added importance for both contagion and amplification once we
introduce bankruptcy costs and mark-to-market reductions in credit quality. Bankruptcy costs steepen
the losses at defaulted nodes, thereby increasing the likelihood that defaults will spread to other nodes.
These losses are further amplified by feedback effects, thus increasing the system-wide loss in value. By
contrast, reductions in credit quality have the effect of marking down asset values in advance of default.
This process is akin to a slippery slope: once some node suffers a deterioration in its balance sheet,
its mark-to-market value decreases, which reduces the value of the nodes to which it has obligations,
causing their balance sheets to deteriorate. The result can be a system-wide reduction in value that was
triggered solely by a loss of confidence rather than an actual default.
References
[1] Acemoglu, D., Ozdaglar, A., and Tahbaz-Salehi, A. (2013). Systemic Risk and Stability in Finan-
cial Networks. Working Paper 18727, National Bureau of Economic Research, Cambridge, Mas-
sachusetts.
[2] Allen, F., and Babus, A. (2009). Networks in Finance. In The Network Challenge: Strategy, Profit,
and Risk in an Interlinked World (eds. P. Kleindorfer, Y. Wind & R. Gunther), Wharton School
Publishing, Upper Saddle River, New Jersey.
[3] Allen, F., and Carletti, E. (2011). New Theories to Underpin Financial Reform, Journal of Financial
Stability, in press.
[4] Allen, F., and Gale, G. (2000). Financial Contagion, Journal of Political Economy 108 1–33.
28
[5] Bank of England (2011). Financial Stability Report 29. Available at www.bankofengland.co.uk.
[6] Barlow, R.E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing, Holt,
Rinehart, and Winston, New York.
[7] Battiston, S., Delli Gatti, D., Gallegati, M., Greenwald, B., and Stiglitz, J.E. (2012). Default
Cascades: When Does Risk Diversification Increase Stability? Journal of Financial Stability 8,
138-149.
[8] Caccioli, F., Shrestha, M., Moore, C., and Farmer, J.D. (2012). Stability Analysis of Financial
Contagion Due to Overlapping Portfolios. Working Paper 2012-10-018, Santa Fe Institute.
[9] Cifuentes, R., Ferrucci, G., and Shin, H.S. (2005). Liquidity Risk and Contagion, Journal of the
European Economic Association 3, 556–566.
[10] Cont, R., Moussa, A., and Santos, E. (2010). Network Structure and Systemic Risk in Banking
Systems. Available at SSRN: http://ssrn.com/abstract=1733528.
[11] Degryse, H., and Nguyen, G. (2004). Interbank Exposures: An Empirical Examination of Systemic
Risk in the Belgian Banking System. Working paper, Belgian National Bank.
[12] Demange, G. (2012). Contagion in Financial Networks: A Threat Index. Working Paper 2012-02,
Paris School of Economics, Paris.
[13] Diamond, D.W., and Dybvig, P. (1983). Bank Runs, Deposit Insurance, and Liquidity, Journal of
Political Economy 91, 401–419.
[14] Eisenberg, L., and Noe, T.H. (2001). Systemic Risk in Financial Systems, Management Science 47,
236–249.
[15] Elliott, M., Golub, B., and Jackson, M.O. (2013) Financial Networks and Contagion. Working
Paper, California Institute of Technology, Pasadena, California.
[16] Elsinger, H., Lehar, A., and Summer, M. (2006). Risk Assessment for Banking Systems, Management
Science 52, 1301–1314.
[17] European Banking Authority (2011), EU-Wide Stress Test Results. Available at www.eba.europa.eu.
[18] Furfine, C. (2003). Interbank Exposures: Quantifying the Risk of Contagion, Journal of Money,
Credit and Banking 35, 11–128.
[19] Gai, P., Haldane, A., and Kapadia, S. (2011). Complexity, Concentration, and Contagion, Journal
of Monetary Economics 58, 453–470.
29
[20] Gai, P., and Kapadia, S. (2010). Contagion in Financial Networks, Proceedings of the Royal Society
A 466, 2401-2423.
[21] Georg, C.-P. (2011). The Effect of the Interbank Network Structure on Contagion and Common
Shocks, Deutsche Bundesbank Discussion Paper Series 2: Banking and Financial Studies, No.
12/2011.
[22] Goodhart, C.A.E., Sunirand, P., and Tsomocos, D. P. (2004). A Model to Analyse Financial
Fragility: Applications, Journal of Financial Stability 1, 1-30.
[23] Haldane, A. G., and May, R. M. (2011). Systemic Risk in Banking Ecosystems, Nature 469, 351-355.
[24] Harris, T. S., Herz, R.H., and Nissim, D. (2012) Accounting’s Role in the Reporting, Creation,
and Avoidance of Systemic Risk in Financial Institutions. In The Handbook of Systemic Risk (eds.
J.P-Fouqe and J. Langsam), Cambridge University Press.
[25] International Monetary Fund (2012). Global Financial Stability Report: Restoring Confidence and
Progressing on Reforms. Available at www.imf.org/external/pubs/ft/gfsr.
[26] Liu, M., and Staum, J. (2012). Systemic Risk Components in a Network Model of Contagion.
Available at SSRN: http://ssrn.com/abstract=1726107.
[27] Marshall, A.W., and Olkin, I. (1974). Majorization in Multivariate Distributions. The Annals of
Statistics 2, 1189-1200.
[28] Morris, S. (2000). Contagion. Review of Economic Studies 67, 57-78.
[29] Newman, M. (2010). Networks: An Introduction. Oxford University Press, USA.
[30] Nier, E., Yang, J., Yorulmazer, T., and Alentorn, A. (2007) Network Models and Financial Stability,
Journal of Economic Dynamics and Control 31, 2033-2060.
[31] Office of Financial Research (2012). Annual Report. Available at www.treasury.gov/ofr.
[32] Ramos Romer, H.M., and Sordo Diaz, M.A. (2001) The Proportional Likelihood Ratio Order and
Applications. QUESTIIO 25, 211-223.
[33] Rogers, L.C.G., and Veraart, L.A.M. (2012). Failure and Rescue in an Interbank Network. Manage-
ment Science, in press.
[34] Shin, H. (2012) Global Banking Glut and Loan Risk Premium. Mundell-Fleming Lecture, Interna-
tional Monetary Fund, Washington, D.C.
30
[35] Shleifer, A., and Vishny, R. (2011). Fire Sales in Finance and Economics. Journal of Economic
Perspectives 25, 29–48.
[36] Tasche, D. (2008). The Vasicek Distribution. Available at wwwold-m4.ma.tum.de/pers/tasche/.
[37] Upper, C., (2011). Simulation Methods to Assess the Danger of Contagion in Interbank Markets,
Journal of Financial Stability 7, 111-125.
[38] Upper, C., and Worms, A. (2002). Estimating Bilateral Exposures in the German Interbank Market:
Is There a Danger of Contagion? Discussion Paper 09, Deutsche Bundesbank.
[39] Vuillemey, G., and Peltonen, T.A. (2013) Sovereign Credit Events and Their Spillovers to the
European Banking System — The Interplay Between Sovereign Bonds and CDS Holdings. Working
paper, European Central Bank.
[40] Wirch, J.L. (1999) Coherent Beta Risk Measures for Capital Requirements. Dissertation, Depart-
ment of Statistics and Actuarial Sciences, University of Waterloo, Waterloo, Ontario, Canada.
[41] Yellen, J. (2013) Interconnectedness and Systemic Risk: Lessons from the Financial Crisis and
Policy Implications. Board of Governors of the Federal Reserve System, Washington, D.C.
31
Appendix: Application to European Banks
To provide some insight into how our theoretical results apply in practice, we draw on data from banks
that participated in the European Banking Authority’s (EBA) 2011 stress test. Detailed information on
interbank exposures needed to calibrate a full network model is not publicly available. But our results
do not depend on detailed network structure, so the information disclosed with the results of the stress
test give us most of what we need.
Ninety banks in 21 countries participated in the stress test. For each, the EBA reports total assets
and equity values as of the end of 2010. In addition, the EBA reports each bank’s total exposure at
default (EAD) to other financial institutions. The EAD measures a bank’s total claims on all other
banks, so we take this as the size of each bank’s in-network assets. Subtracting this value from total
assets gives us ci, the size of the bank’s outside assets. For wi we use the equity values reported by the
EBA, which then allows us to calculate λi = ci/wi.
The only remaining parameter we need is βi, the fraction of a bank’s liabilities owed to other banks.
This information is not included in the EBA summary, nor is it consistently reported by banks in their
financial statements. As a rough indication, we assume that each bank’s in-network liabilities equal its
in-network assets (though we will see that our results are fairly robust to this assumption).14 This gives
us βi = EAD/(assets-equity).
Some of the smallest banks have problematic data, so as a simple rule we omit the ten smallest. We
also omit any countries with only a single participating bank. This leaves us with 76 banks, of which
the 50 largest are included in Table 1. For all 76 banks, Table 2 includes assets, EAD, and equity as
reported by the EBA (in millions of euros), and our derived values for ci, wi, βi, λi. The banks are listed
by asset size in Table 1 and grouped by country in Table 2.
In Table 1, we examine the potential for contagion from the failure of each of the five largest banks,
BNP Paribas, Deutsche Bank, HSBC, Barclays, and Credit Agricole. Taking each of these in turn as
the triggering bank, we then take the default set D to be consecutive pairs of banks. The first default
set under BNP Paribas consists of Deutsche Bank and HSBC, the next default set consists of HSBC and
Barclays, and so on.
Under “WR” (weak ratio), we report the ratio of the left side of inequality (20) to the right side.
Contagion is weak whenever this ratio exceeds 1, as it does in most cases in the table. Contagion fails to
be weak only when the banks in the default set are much smaller than the triggering bank. Moreover,
the ratio reported for each bank shows how much greater βi would have to be to reverse the direction
of inequality (20). For example, the first ratio listed under BNP Paribas, corresponding to the default
14Based on Federal Reserve Release H.8, the average value of βi for commercial banks in the U.S. is about 3%, so ourestimates for European banks would appear to be conservative.
32
of Deutsche Bank and HSBC, is 18.64, based on a βi value of 4.6% (see Table 2). This tells us that
the βi value would have to be at least 18.64 x 4.6% = 85.7% for the weak contagion condition to be
violated. In this sense, the overall pattern in Table 1 is robust to our estimated values of βi. Expanding
the default sets generally makes contagion weaker because of the relative magnitudes of wi and λ−1i ; see
(20) and Table 2.
Under “LR” (for likelihood ratio), we report the relative probability of failure through independent
direct shocks and through contagion, calculated as the ratio of the right side of (22) to the left side.
This is the ratio of probabilities under the assumption of a uniform distribution p = q = 1, which
is conservative. An asterisk indicates that the ratio is infinite because default through contagion is
impossible.15 The value of 6.68 reported under BNP Paribas for the default set consisting of Banco
Santander and Societe Generale indicates that the probability of default through independent shocks is
6.68 times more likely than default through contagion. Raising the LR values in the table to a power of
q > 1 gives the corresponding ratio of probabilities under a shock distribution having parameters p = 1
and q.16
Table 2 reports a similar analysis by country. Within each country, we first consider the possibility
that the failure of a bank causes the next two largest banks to default – the next two largest banks
constitute the default set D. (In the case of Belgium, there is only one bank to include in D.) For each
bank, the column labeled “Ratio for Next Two Banks” reports the ratio of the left side of inequality
(20) to the right side. As in Table 1, contagion is weak whenever this ratio exceeds 1. The table shows
that the ratio is greater than 1 in every case. As in Table 1, the magnitude of the ratio also tells us how
much larger βi would have to be to reverse the inequality in (20).
The last column of the table reports the corresponding test for weak contagion but now holding D
fixed as the two smallest banks in each country group. We now see several cases in which the ratio is
less than 1 – for example, when we take Deutsche Bank to be the triggering bank and the two smallest
banks in the German group (Landesbank Berlin and DekaBank Deutsch Girozentrale) as the default set
we get a ratio of 0.5 in the last column.
The results in Tables 1 and 2 reflect the observations we made following Theorem 1 about the relative
magnitudes needed for the various model parameters in order that contagion not be weak, and they show
that the parameter ranges implying weak contagion are indeed meaningful in practice.
15In deriving (32), we use the bound in (38), which is conservative. Thus, LR must be greater than 1 whenever WR is,and WR can be finite even when LR is infinite and default through contagion is impossible.
16We noted previously that a beta distribution can be used to approximate the Gaussian copula loss distribution usedin Basel capital standards. We find that the best fit occurs at values of q around 20 or larger. Raising the likelihood ratiosin the table to the qth power thus has a major impact.
33
WR LR WR LR WR LR WR LR WR LR
BNP PARIBAS
DEUTSCHE BANK AG
HSBC HOLDINGS plc 18.64 *
BARCLAYS plc 18.63 * 9.21 3.89
CREDIT AGRICOLE 17.83 * 8.81 2.01 8.27 1.88 27.23
BANCO SANTANDER S.A. 14.91 * 7.37 1.98 6.92 1.85 24.62 *
SOCIETE GENERALE 12.52 6.68 6.19 1.62 5.81 1.55 20.67 * 13.66 15.32
LLOYDS BANKING GROUP plc 11.23 7.70 5.55 1.63 5.21 1.56 18.55 * 12.26 23.52
BPCE 11.29 12.77 5.58 1.70 5.24 1.62 18.65 * 12.32 *
ING BANK NV 10.32 3.44 5.10 1.46 4.79 1.41 17.05 * 11.27 4.57
UNICREDIT S.p.A 9.55 4.06 4.72 1.49 4.43 1.44 15.77 * 10.42 5.90
COMMERZBANK AG 8.47 3.33 4.18 1.43 3.93 1.38 13.98 * 9.24 4.39
RABOBANK NEDERLAND 6.96 2.48 3.44 1.33 3.23 1.29 11.49 * 7.59 2.94
ROYAL BANK OF SCOTLAND GROUP plc 6.06 * 2.99 1.67 2.81 1.58 10.01 * 6.61 *
INTESA SANPAOLO S.p.A 5.70 72.67 2.82 1.63 2.64 1.54 9.41 * 6.22 *
DEXIA 4.59 1.79 2.27 1.19 2.13 1.16 7.58 5.19 5.01 1.97
Nordea Bank AB (publ) 4.52 1.57 2.23 1.15 2.10 1.13 7.46 2.96 4.93 1.68
BANCO BILBAO VIZCAYA ARGENTARIA 5.24 1.85 2.59 1.21 2.43 1.19 8.65 5.82 5.72 2.05
DANSKE BANK 4.47 1.66 2.21 1.16 2.08 1.14 7.39 3.72 4.88 1.80
ABN AMRO BANK NV 3.91 1.33 1.93 1.09 1.81 1.08 6.46 1.85 4.27 1.38
Landesbank Baden‐Württemberg 3.36 1.23 1.66 1.06 1.56 1.05 5.55 1.57 3.67 1.27
Hypo Real Estate Holding AG, Münche 3.00 1.15 1.48 1.03 1.39 1.03 4.96 1.34 3.28 1.17
BFA‐BANKIA 3.38 1.21 1.67 1.05 1.57 1.04 5.59 1.49 3.69 1.24
DZ BANK AG Dt. Zentral‐Genossenscha 2.83 1.21 1.40 1.04 1.31 1.03 4.67 1.54 3.08 1.25
Bayerische Landesbank 2.39 1.16 1.18 1.02 1.11 1.01 3.94 1.42 2.61 1.20
KBC BANK 2.73 1.24 1.35 1.04 1.27 1.03 4.50 1.63 2.98 1.28
CAJA DE AHORROS Y PENSIONES DE BA 3.03 1.24 1.50 1.05 1.41 1.04 5.01 1.63 3.31 1.29
BANCA MONTE DEI PASCHI DI SIENA S 2.96 1.17 1.46 1.04 1.38 1.03 4.89 1.40 3.23 1.20
Svenska Handelsbanken AB (publ) 2.61 1.13 1.29 1.02 1.21 1.02 4.31 1.30 2.85 1.15
Norddeutsche Landesbank ‐GZ 2.13 1.09 1.05 1.00 0.99 1.00 3.52 1.22 2.33 1.11
Skandinaviska Enskilda Banken AB (pub 1.96 1.09 0.97 1.00 0.91 0.99 3.24 1.25 2.14 1.11
DnB NOR Bank ASA 2.26 1.16 1.12 1.02 1.05 1.01 3.73 1.44 2.46 1.20
Erste Bank Group (EBG) 2.22 1.17 1.10 1.01 1.03 1.01 3.66 1.47 2.42 1.21
WestLB AG, Düsseldorf 1.91 1.10 0.94 1.00 0.88 0.99 3.15 1.27 2.08 1.12
Swedbank AB (publ) 1.82 1.07 0.90 0.99 0.84 0.99 3.00 1.19 1.98 1.09
Nykredit 1.98 1.10 0.98 1.00 0.92 0.99 3.27 1.26 2.16 1.12
BANK OF IRELAND 1.76 1.08 0.87 0.99 0.82 0.98 2.91 1.24 1.92 1.10
HSH Nordbank AG, Hamburg 1.63 1.06 0.80 0.98 0.76 0.98 2.69 1.18 1.78 1.08
BANCO POPOLARE ‐ S.C. 1.58 1.05 0.78 0.98 0.73 0.98 2.61 1.15 1.73 1.06
Landesbank Berlin AG 1.21 1.03 0.60 0.96 0.56 0.95 1.99 1.13 1.32 1.04
ALLIED IRISH BANKS PLC 1.12 1.01 0.55 0.96 0.52 0.96 1.85 1.10 1.22 1.03
Raiffeisen Bank International (RBI) 1.23 1.03 0.61 0.96 0.57 0.95 2.04 1.14 1.35 1.05
UNIONE DI BANCHE ITALIANE SCPA (U 1.22 1.04 0.60 0.95 0.57 0.94 2.02 1.19 1.33 1.06
DekaBank Deutsche Girozentrale, Fran 1.19 1.02 0.59 0.96 0.55 0.96 1.96 1.12 1.30 1.04
BANCO POPULAR ESPAÑOL, S.A. 1.21 1.03 0.60 0.96 0.56 0.96 2.00 1.12 1.32 1.04
CAIXA GERAL DE DEPÓSITOS, SA 1.27 1.04 0.63 0.96 0.59 0.95 2.10 1.18 1.39 1.06
NATIONAL BANK OF GREECE 1.25 1.05 0.62 0.95 0.58 0.94 2.07 1.21 1.37 1.07
BANCO COMERCIAL PORTUGUÊS, SA (B 1.21 1.03 0.60 0.95 0.56 0.95 2.00 1.15 1.32 1.05
BANCO DE SABADELL, S.A. 1.08 1.01 0.53 0.96 0.50 0.96 1.78 1.07 1.17 1.02
EFG EUROBANK ERGASIAS S.A. 1.01 1.00 0.50 0.95 0.47 0.95 1.66 1.07 1.10 1.01
ESPÍRITO SANTO FINANCIAL GROUP, SA 0.92 0.99 0.46 0.94 0.43 0.94 1.52 1.07 1.01 1.00
From BNP Paribas From Deutsche Bank From HSBC From Barclays From Credit Agricole
Table 1: Results based on 2011 EBA stress test data. We consider contagion from each of the five largest banksacross the top to pairs of banks in consecutive rows. A weak ratio (WR) value greater than 1 indicates thatcontagion is weak. Each LR value is a likelihood ratio for default through independent shocks to default throughcontagion, assuming a uniform fractional shock.
34
c_i w_i beta_i lambda_i
Assets EAD Outside Assets Equity
EAD/(Assets‐
Equity)
Outside
Assets/Equity
Ratio for Next
Two Banks
Ratio for Last
Two Banks
AT001 Erste Bank Group (EBG) 205,938 25,044 180,894 10,507 12.8% 17.2
AT002 Raiffeisen Bank International (RBI) 131,173 30,361 100,812 7,641 24.6% 13.2
AT003 Oesterreichische Volksbank AG 44,745 10,788 33,957 1,765 25.1% 19.2 3.4
BE004 DEXIA 548,135 228,211 319,924 17,002 43.0% 18.8
BE005 KBC BANK 276,723 23,871 252,852 11,705 9.0% 21.6 35.7
CY006 MARFIN POPULAR BANK PUBLIC CO LTD 42,580 7,907 34,673 2,015 19.5% 17.2
CY007 BANK OF CYPRUS PUBLIC CO LTD 41,996 7,294 34,702 2,134 18.3% 16.3 1.6
DE017 DEUTSCHE BANK AG 1,905,630 194,399 1,711,231 30,361 10.4% 56.4 0.5
DE018 COMMERZBANK AG 771,201 138,190 633,011 26,728 18.6% 23.7 0.8
DE019 Landesbank Baden‐Württemberg 374,413 133,906 240,507 9,838 36.7% 24.4 2.5 1.0
DE020 DZ BANK AG Dt. Zentral‐Genossenschaf 323,578 135,860 187,718 7,299 43.0% 25.7 1.9 1.1
DE021 Bayerische Landesbank 316,354 97,336 219,018 11,501 31.9% 19.0 2.4 1.3
DE022 Norddeutsche Landesbank ‐GZ 228,586 91,217 137,369 3,974 40.6% 34.6 2.5 1.6
DE023 Hypo Real Estate Holding AG, München 328,119 29,084 299,035 5,539 9.0% 54.0 3.0 3.3
DE024 WestLB AG, Düsseldorf 191,523 58,128 133,395 4,218 31.0% 31.6 3.6 2.2
DE025 HSH Nordbank AG, Hamburg 150,930 9,532 141,398 4,434 6.5% 31.9 5.2 9.7
DE027 Landesbank Berlin AG 133,861 49,253 84,608 5,162 38.3% 16.4 2.6
DE028 DekaBank Deutsche Girozentrale, Frank 130,304 41,255 89,049 3,359 32.5% 26.5 9.7
DK008 DANSKE BANK 402,555 75,894 326,661 14,576 19.6% 22.4 0.4
DK011 Nykredit 175,888 8,597 167,291 6,633 5.1% 25.2 2.7
DK009 Jyske Bank 32,752 4,674 28,078 1,699 15.1% 16.5 1.4
DK010 Sydbank 20,238 3,670 16,568 1,231 19.3% 13.5 2.7
ES059 BANCO SANTANDER S.A. 1,223,267 51,407 1,171,860 41,998 4.4% 27.9 0.4
ES060 BANCO BILBAO VIZCAYA ARGENTARIA S 540,936 110,474 430,462 24,939 21.4% 17.3 0.2
ES061 BFA‐BANKIA 327,930 39,517 288,414 13,864 12.6% 20.8 7.4 0.6
ES062 CAJA DE AHORROS Y PENSIONES DE BA 275,856 5,510 270,346 11,109 2.1% 24.3 3.2 3.6
ES064 BANCO POPULAR ESPAÑOL, S.A. 129,183 14,810 114,373 6,699 12.1% 17.1 5.2 1.5
ES065 BANCO DE SABADELL, S.A. 96,703 3,678 93,025 3,507 3.9% 26.5 19.6 5.5
ES066 CAIXA D'ESTALVIS DE CATALUNYA, TAR 76,014 8,219 67,795 3,104 11.3% 21.8 6.1 2.7
ES067 CAIXA DE AFORROS DE GALICIA, VIGO, 73,319 2,948 70,371 2,849 4.2% 24.7 19.5 6.9
ES083 CAJA DE AHORROS DEL MEDITERRANEO 72,034 4,981 67,053 1,843 7.1% 36.4 9.5 4.2
ES071 GRUPO BANCA CIVICA 71,055 7,419 63,636 3,688 11.0% 17.3 22.9 3.0
ES068 GRUPO BMN 69,760 7,660 62,101 3,304 11.5% 18.8 13.6 2.9
ES063 EFFIBANK 54,523 4,124 50,399 2,656 8.0% 19.0 8.5 5.2
ES069 BANKINTER, S.A. 53,476 2,141 51,335 1,920 4.2% 26.7 7.5 9.5
ES070 CAJA ESPAÑA DE INVERSIONES, SALAM 45,656 7,235 38,422 2,076 16.6% 18.5 11.5 3.2
ES075 GRUPO BBK 44,628 1,924 42,704 2,982 4.6% 14.3 19.9 10.7
ES072 CAJA DE AHORROS Y M.P. DE ZARAGOZ 42,716 1,779 40,938 2,299 4.4% 17.8 6.9 11.5
ES073 MONTE DE PIEDAD Y CAJA DE AHORRO 34,263 2,599 31,664 2,501 8.2% 12.7 19.4 8.2
ES074 BANCO PASTOR, S.A. 31,135 1,665 29,470 1,395 5.6% 21.1 18.1 12.5
ES076 CAIXA D'ESTALVIS UNIO DE CAIXES DE M 28,310 1,802 26,508 1,065 6.6% 24.9 11.8 11.6
ES077 CAJA DE AHORROS Y M.P. DE GIPUZKOA 20,786 285 20,501 1,935 1.5% 10.6 14.2
ES078 GRUPO CAJA3 20,144 1,926 18,218 1,164 10.1% 15.7 11.6
FR013 BNP PARIBAS 1,998,157 90,328 1,907,829 55,352 4.6% 34.5 11.1
FR014 CREDIT AGRICOLE 1,503,621 83,713 1,419,908 46,277 5.7% 30.7 12.2
FR016 SOCIETE GENERALE 1,051,323 100,013 951,310 27,824 9.8% 34.2 13.9
FR015 BPCE 1,000,695 34,983 965,712 31,943 3.6% 30.2 12.2
GB089 HSBC HOLDINGS plc 1,783,199 212,092 1,571,107 86,900 12.5% 18.1 3.5
GB090 BARCLAYS plc 1,725,709 53,873 1,671,836 46,232 3.2% 36.2 12.3
GB091 LLOYDS BANKING GROUP plc 1,006,082 29,233 976,849 47,984 3.1% 20.4 6.6
GB088 ROYAL BANK OF SCOTLAND GROUP plc 607,351 105,506 501,845 58,982 19.2% 8.5 12.3
GR031 NATIONAL BANK OF GREECE 118,832 8,608 110,224 8,153 7.8% 13.5 6.0
GR030 EFG EUROBANK ERGASIAS S.A. 85,885 3,838 82,047 4,296 4.7% 19.1 12.9
GR032 ALPHA BANK 66,798 3,492 63,306 5,275 5.7% 12.0 8.9 14.4
GR033 PIRAEUS BANK GROUP 57,680 1,581 56,099 3,039 2.9% 18.5 16.5
GR034 AGRICULTURAL BANK OF GREECE S.A. ( 31,221 1,657 29,564 792 5.4% 37.3 14.4
IE038 BANK OF IRELAND 156,712 17,254 139,458 7,037 11.5% 19.8 4.9
IE037 ALLIED IRISH BANKS PLC 131,311 11,277 120,034 3,669 8.8% 32.7
IE039 IRISH LIFE AND PERMANENT 46,743 6,127 40,616 1,681 13.6% 24.2 4.9
IT041 UNICREDIT S.p.A 929,488 106,707 822,781 35,702 11.9% 23.0 1.3
IT040 INTESA SANPAOLO S.p.A 576,962 109,909 467,053 26,159 20.0% 17.9 1.4
IT042 BANCA MONTE DEI PASCHI DI SIENA S.p 244,279 12,074 232,205 6,301 5.1% 36.9 4.2 10.4
IT043 BANCO POPOLARE ‐ S.C. 140,043 7,602 132,440 5,474 5.6% 24.2 2.0
IT044 UNIONE DI BANCHE ITALIANE SCPA (UB 130,559 19,793 110,766 6,559 16.0% 16.9 10.4
NL047 ING BANK NV 933,073 111,756 821,317 30,895 12.0% 26.6 2.5
NL048 RABOBANK NEDERLAND 607,483 37,538 569,945 27,725 6.2% 20.6 7.2
NL049 ABN AMRO BANK NV 379,599 29,196 350,403 11,574 7.7% 30.3 5.1
NL050 SNS BANK NV 78,918 388 78,530 1,782 0.5% 44.1 7.2
PT053 CAIXA GERAL DE DEPÓSITOS, SA 119,318 14,221 105,097 6,510 11.9% 16.1 5.0
PT054 BANCO COMERCIAL PORTUGUÊS, SA (B 100,010 7,690 92,320 3,521 7.7% 26.2 8.5
PT055 ESPÍRITO SANTO FINANCIAL GROUP, SA 85,644 8,690 76,955 4,520 10.1% 17.0 7.1
PT056 Banco BPI, SA 43,826 5,463 38,364 2,133 12.5% 18.0 8.5
SE084 Nordea Bank AB (publ) 542,853 61,448 481,405 19,103 11.3% 25.2 3.7
SE085 Skandinaviska Enskilda Banken AB (pub 212,240 25,955 186,284 9,604 12.2% 19.4 9.0
SE086 Svenska Handelsbanken AB (publ) 240,202 20,870 219,332 8,209 8.7% 26.7 3.8
SE087 Swedbank AB (publ) 191,365 17,358 174,007 7,352 9.1% 23.7 9.0
Bank Number and Name
Table 2: Results based on 2011 EBA stress test data. In the last two columns, a ratio greater than 1 indicatesweak contagion from a bank to the next two largest banks in the same country and to the two smallest banks inthe country group, respectively. 35